We will discuss the fluctuations in discrete Beta-ensembles. Such distributions describe for instance the positions of lozenge tilings and ressemble the continuous Beta-ensembles which appear in random matrix ensembles. We will highlight the role of equations introduced by Nekrasov in this context, which play the same crucial role than Dyson-Schwinger equations in the continuous case, but are not obtained by a straightforward integration by part.

I will discuss the asymptotics of the spectrum of non-normal random matrices, whose law is invariant under the action of the orthogonal or the unitary group. This is based on a joint work with M. Krishnapur and O. Zeitouni.

I will discuss the construction of matrix models related with the combinatorics of loop models, based on ideas coming from Jones planar algebra theory. As an application, I will give an exact solution for the Potts model on random graphs, based on random matrices techniques. This is based on joint works with V. Jones, D. Shlyakhtenko and P. Zinn Justin.